THE METHOD OF CONTROLLED LAGRANGIANS: ENERGY PLUS FORCE SHAPING

Abstract

We study the method of controlled Lagrangians to stabilize mechanical systems connected with external forces. The basic idea is that we transform by feedback a given controlled Lagrangian system to another controlled Lagrangian system with positive definite energy and a dissipative external force such that a dissipative feedback force stabilizes the closed-loop system. We derive matching conditions for energy plus force shaping that are more general and stronger than those in the literature. We provide various easily verifiable criteria for stabilizability by the method of controlled Lagrangians, including a necessary and sufficient condition for stabilizability by energy shaping for all linear mechanical systems, a necessary and sufficient condition for stabilizability by energy shaping for the class of all mechanical systems with one degree of underactuation, and a sufficient condition for stabilizability by energy plus force shaping for the class of all mechanical systems with one degree of underactuation connected with external forces.